"In normal operation the PicoScope 320x is powered either from a USB port on your computer or from a powered hub. It is not possible to power this device from an unpowered USB hub. If both the PC and the device under test are referenced to the same ground (earth) then noise/offset problems may occur. This problem is only likely to occur when using a mains powered desktop PC whilst measuring DC coupled (earth referenced) signals of a small amplitude. If this situation occurs a mains power adaptor (supplied) will have to be used to power the unit."

The scope is protected against overload / misconnection by self resetting fuses. Like most USB scopes powered from the PC an earth loop can be created in the above circumstances. Powering instead from the external mains power supply avoids the earth loop as the power now comes from a floating supply.

Thank you for clearing this up. I would put it this way (in technical terms):
The external power supply eliminates the DC voltage drop in the ground wire of the USB cable, caused by the supply current. This can be a problem, especially because the supply current changes over time.

I would also be very grateful if someone could provide more detailed specifications of these self resetting fuses (something like the description
of their operation and connection diagram). It would be nice to know beforehand, when and why the box will kick me in the face.